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<h2 align="center"><font color="#0000FF">Partial Inversion of a Map: Berz's 
DAPIN</font></h2>

<p>Generally a <b><font color="#FF0000">canonical</font></b>&nbsp; DAMAP, i.e., 
a Taylor map centered around (0,0..,0.0) can be expressed as a generating 
function.<!--[if gte mso 9]> We start with a generic DAMAP.<![endif]--></p>
<p align="center"><font size="6">&nbsp;
<img border="0" src="big_ta2.gif" width="299" height="103"></font></p>
<p>To produce a generating function g such that:</p>
<p align="center"><img border="0" src="big_ta6.gif" width="501" height="103"></p>
<p align="left">We must produce equation (2). Notice that equation (2) is 
obtained by inverting the roles of p and p<sup>final</sup>.</p>
<p align="left">We provide a simple nonlinear example of this: click <b>
<a href="../simple_programs/z_damaps_partial_inversion.f90" style="text-decoration: none">
<font color="#FF0000">here</font></a></b>. The linear part of this problem is 
worked out below. It is a rotation of angle Arccos(0.6). </p>
<dl>
  <dd>
<p align="center"><font size="6">M<sub>1 </sub>=&nbsp; 0.6 x + 0.8 p</font></p>
  </dd>
  <dd>
<p align="center"><font size="6">M<sub>2 </sub>= -0.8x + 0.6p</font></p>
  </dd>
</dl>
<p align="left">The partially inverted map is easily computed to be:</p>
<dl>
  <dd>
<p align="center"><font size="6">G<sub>1 </sub>=&nbsp; 5/3 x + 4/3 p</font><sup><font size="6">final</font></sup></p>
  </dd>
  <dd>
<p align="center"><font size="6">G<sub>2 </sub>=&nbsp; 4/3x + 5/3 p</font><sup><font size="6">final</font></sup></p>
  </dd>
</dl>
<div align="left">
  The generating function g is thus:</div>
<div align="left">
&nbsp;</div>
<dl>
  <dd>
<p align="center"><font size="6">g<sub> </sub>=&nbsp; 2/3x<b><sup>2</sup></b> + 
5/3 x p</font><font size="6"><sup>final </sup>+ 2/3 p<sup>final&nbsp; <b>2 </b>
</sup></font></p>
  </dd>
</dl>
<p align="left">&nbsp;</p>
<p align="center"><b><font color="#FF0000" size="6">
<a href="../simple_programs/z_damaps_partial_inversion.f90" style="text-decoration: none">
<font color="#FF0000">Example</font></a></font></b></p>
<p align="left">&nbsp;</p>

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